Help with a special function

[Fred Wright]

While studying the solution to a integral problem I found online I ran across a special function I am unfamiliar with. The integral is
$$
\int_0^{\infty}\frac{t^{\frac{m+1}{n}-1}}{1+t}dt=\mathcal{B}(\frac{m+1}{n},1-\frac{m+1}{n})
$$
This certainly isn't the normal beta function. What is it? Can anyone direct me to a reference on this function?

 

[fresh_42]

I get the ordinary beta function (3rd formula with your exponents):
https://de.wikipedia.org/wiki/Eulersche_Betafunktion
or in the English version further down:
https://en.wikipedia.org/wiki/Beta_function#Other_identities_and_formulas

 

[Fred Wright]

I get the ordinary beta function (3rd formula with your exponents):
https://de.wikipedia.org/wiki/Eulersche_Betafunktion
or in the English version further down:
https://en.wikipedia.org/wiki/Beta_function#Other_identities_and_formulas

Thanks, "Who can read has a clear advantage"

 

[dextercioby]

It's just the "normal Euler's Beta function" in the RHS but with "intimidating arguments" . One trick is to simply replace $\frac{m+1}{n} \equiv \alpha$ in your exponent under the integral sign, do the integral and revert to $m,n$ at the end.

 

[pasmith]

$$\int_0^1 t^{p-1}(1-t)^{q-1}\,dt = \int_0^\infty \frac{z^{p-1}}{(z + 1)^{p+q}}\,dz$$ where $z = t/(1-t)$.